I have heard of some of these terms, but there may be others on this list that I am unaware of, so please edit.

I am uncertain what these terms mean. More curious, why are each of these important, practically? If you can include some history / background to when the grade issue occurred and / or some sources, I appreciate that.

3 Answers
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Tefillin are divided into different categories based on the quality of the leather boxes

Tfilin crafted from two separate pieces of leather (which are then glued together) are known as tfilin peshutim, the simplest tfilin. Hasofer says they generally last only three to five years, after which the pieces begin to separate and they lose their required square shape making them invalid for use

Tfilin made from one very thin piece of leather are called tfilin dakkot, thin tfilin. They are made from a single piece of high grade but thinner leather. The required perfectly square tefillin shape is produced by folding and gluing the leather, or in some cases by stretching the leather over a frame and folding it into shape

Tfilin formed from one very thick piece of leather are called tfilin gassot, thick tfilin. These are the most durable (and expensive) kind. It is more difficult to make and only became available in the last century or so. The leather is stretched over metal plates and folded and pressed at very high-pressure to form the familiar perfectly square tefillin shape. They will last up to 50 years when properly maintained but are of course more expensive

Mehudar (as in pshutim mehudarim or dakot mehudarim) refer to more hiddurim (enhancements)

Prudot (separated) refer to the way the four compartments of the shel rosh are separated. Most tefilin gassot are rov prudot, which means they are separate most of the way down, but some glue is applied toward the bottom to help the tefilin keep their shape over time. Tefillin Prudot are completely separated, to meet the stricter opinions. However, note that in some cases they do not retain their shape and have to be reshaped after a few years. Standard tefillin gassot and prudot look almost identical on the outside

The reason the quality levels are important is simply than the higher-quality will maintain their shape (and kashrut level) much longer. Unless one is on a strict budget, there is no reason today not to buy tfilin gassot. Even economically, it is much cheaper to buy high quality tfilin and keep them 50 years than cheaper ones that need regular fixing/replacement.

The "technology" used to manufacture gassot has only been developed at scale in the last century which is why the tfilin we wear today are of incomparably better quality than those worn by the greatest sages of previous centuries. A wise rav once told me this was because, as we were descending spiritually, maybe we need better religious objects to compensate...

See here, here and here for more sources which I partly leveraged to assemble the definitions above

And here is the best book I found on the intricacies of making tfilin and stam in general. Fascinating reading.

These different terms describing tefillin are not all on the same "plane" so to speak. In other words, they describe different aspects of the tefillin.

In one plane is the spectrum of peshutim, dakot, gasot. These words describe the batim (leather boxes) of the tefillin.

Tefillin peshutim are the lowest quality and least expensive. They are composed of multiple pieces of leather glued together. This is not halachically preferable, although still widely considered kosher. Due to their low cost, they are sometimes used as "starter tefillin." I have heard of Chabads giving them out for free to people who are starting along the path of observant Judaism but who aren't yet ready to make the financial commitment to a superior tefillin set.

Dakot are made from one thin piece of leather. This may come from a smaller kosher animal (such as a deer) or from a less thick piece of cow skin. Being made from a single piece of leather, they are halachically superior to peshutim. Because they are thinner and therefore more fragile than gasot they are relatively uncommon now; however, manufacturing processes only advanced to the point of being able to manufacture gasot quite recently and before 100ish years ago, dakot were the standard.

Gasot are made from a much thicker piece of leather than dakot. Working the leather into the shape of tefillin requires huge amounts of pressure from machines. These are the most durable and also the most expensive batim. They are the most common now, though, because while dakot need to be replaced every so often because they lose their shape from accidental banging and repeated use (which makes them pasul), gasot can be expected to last a lifetime (making them a good investment).

Gasot prudot are tefillin gasot whose compartments of the shel rosh reach all the way to the base of the bayit. As far as I know, this is more of a hidur mitzvah than fulfilling a halachic stringency (but I could be wrong about that).

On the other plane is the description mehudarim. This is usually used to refer to the writing of the tefillin rather than the batim, but in reality the word could be used for the batim as well. Mehudar just means "beautiful" (as in hidur mitzvah). Tefillin with beautiful writing are often described as mehudarim, although the word could also in theory be used to describe beautifully made batim. Note that by "beautiful" I do not just mean "nicer to look at." I mean that the aspects that are mehudar beautify the mitzvah. Aspects that beautify the mitzvah include the crisp clarity of the written text, the perfect squareness of the batim, as well as the yirat shamayim of the sofer.

The following material is taken from hasofer.com and may answer most of your questions.

What you are referring to as "Gassos Prudos" sounds like thick skins that are not from a single skin but are glued together. This would be by definition, less mehudar.

Minimally kosher or mehudar? It's your choice!

Jewish law recognizes 4 levels of kashrut. An item that meets the minimum standards of halacha is called kosher l'hatchila l'bracha, kosher or fitting for a person to recite the appropriate blessing and use it to perform a mitzvah. If the item doesn't meet even these standards due to its containing halachically-condoned errors or poor quality it may be considered kosher b'diavad, minimally kosher according to halacha: permissible for use but only when no item of higher quality is available. Pasul refers to an item which doesn't meet even the lowest of standards and which is not suitable for use for a mitzvah.

The highest level of kashrut is mehudar, beautiful or enhanced. The Torah teaches us that in addition to fulfilling the mitzvot we must also beautify them to the best of our abilities. This concept applies equally to external aspects of the mitzvot that we can see as well as to hidden or unknowable ones. The technical expertise and quality of the sofer's craftsmanship as seen in his writing and the perfect square of the tefillin batim are visible to the human eye. But more important are his yirat shamayim, personal integrity, and the breadth and depth of his knowledge of the laws of STa"M, as emphasized by the Chofetz Chaim's words quoted above. Going one step further, the Talmud in tractate Succah teaches us that even the kulmus, the sofer's feather, should be beautiful. Some sofrim store their kulmus in a silver stand for this reason.

It is crucial to know the source of your tefillin and mezuzot, who wrote and examined them, and who put them together. The sofer, examiner, and batim maker must all be G-d fearing individuals, professionally trained and properly certified by recognized Rabbinical authorities.

The two most important components of your tefillin are the parshiot, parchments, and the batim, housings. These determine 90% of the price of your tefillin.

...

HaSOFER tefillin are divided into two types according to the tefillin batim, housings:

Gassot tefillin batim, also called ohr echad, are made of a single piece of thick high grade leather that is stretched over metal plates and folded and pressed to form the familiar perfectly square tefillin shape. These are the very best type of tefillin batim available, both in their adherence to halacha and in their longevity. Exclusively at HaSOFER, the Hebrew letter shin on both sides of the tefillin shel rosh is manually pulled out of the leather while it is still moist, using a unique pliers-like tool we designed ourselves. They are made with a natural drying process that lasts 2 years and have a lifetime of up to 60 years when properly maintained. Gassot tefillin are an excellent long-term investment if it’s within your budget.

Tefillin p'shutim mehudarim dakot batim are also made from a single piece of high grade but thinner leather. The required perfectly square tefillin shape is produced by folding and gluing the leather. Average lifespan of HaSOFER'S tefillin p'shutim mehudarim is seven to ten years, but this can be extended to up to twenty years when properly maintained.

Two other types of tefillin batim, which HaSOFER doesn't carry, are p'shutim and dakot. Tefillin p'shutim are made by gluing together many individual pieces of thin leather. These generally last only three to five years, after which the pieces begin to separate and the tefillin p'shutim lose their required square shape making them invalid for use. Dakot tefillin are made from a single piece of thin leather stretched over a frame and folded into shape. When made in the ideal manner they can last up to twenty years. Dakot tefillin are very difficult to make and are not currently manufactured at a kashrut level that meets the standards of HaSOFER.

I believe you misunderstood gassot prudos - it refers to the separation between the four compartments of the shel rosh - see my answer above
– mblochFeb 2 '16 at 15:11

@mbloch I'm not a Sofer but according to the HaSofer site, when glueing is involved, it is less mehudar. I would have to pull out a couple of reference texts to check, but that is also how I remember it. Two good reference books in English are from Rabbi Shimon Eider, z"l and Rabbi Aryeh Kaplan, z"l. As memory serves, they also said the most mehudar is single skin, without gluing. It's possible it is a regional preference. eichlers.com/…artscroll.com/Books/9781879016064.html
– Yaacov DeaneFeb 2 '16 at 15:23

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This answer is barely on this side of plagiarism. You have two sentences of your own which aren't even correct and then copy-pasted the rest from hasofer.com
– DanielFeb 2 '16 at 15:24

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I think you are confusing single skin vs. multiple (which is the difference between pshutim vs. gassot) and prudot which refers to the way the four compartments are separated (completely/they hold on their own vs. glued). Most poskim hold the glue for prudot is not an issue, while the glue for pshutim is an issue because it will affect the squareness of the bayit. See the definition in my answer. Hope I am clear. Only saying this so you don't leave something incorrect which might confuse others
– mblochFeb 2 '16 at 15:27

@mbloch What HaSofer is calling "Tefillin p'shutim mehudarim" are single skin and have glue to make the batim square, like you describe. As he lists there, they don't last as long and he doesn't place in the category of "the very best type of tefillin batim available".
– Yaacov DeaneFeb 2 '16 at 15:34